Representation theorems are proved for Banach ideal spaces with the Fatou property
which are built by the Calder{\'o}n--Lozanovski\u\i\ construction.
Factorization theorems for operators in spaces more general than the Lebesgue
$L^{p}$ spaces are investigated. It is natural to extend the Gagliardo
theorem on the Schur test and the Rubio de~Francia theorem on factorization of the
Muckenhoupt $A_{p}$ weights to reflexive Orlicz spaces. However, it turns out that for
the scales far from $L^{p}$-spaces this is impossible. For the concrete integral operators
it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces
are not valid. Representation theorems for the Calder{\'o}n--Lozanovski\u\i\ construction
are involved in the proofs.

We present ``reiteration theorems'' with limiting values
$\theta=0$ and $\theta = 1$ for a real interpolation method
involving broken-logarithmic functors. The resulting spaces
lie outside of the original scale of spaces and to describe them
new interpolation functors are introduced. For an ordered couple
of (quasi-) Banach spaces similar results were presented without
proofs by Doktorskii in [D].